Dynamic Programming Problem Set Partial Solution CMPSC 465
|
|
- Brandon Walsh
- 7 years ago
- Views:
Transcription
1 Dynamic Programming Problem Set Partial Solution CMPSC 465 I ve annotated this document with partial solutions to problems written more like a test solution. (I remind you again, though, that a formal problem write-up is not a test. You must introduce everything you re doing. You must write so that your writing tells everything that is going on. That requires good transitions and writing that flows, like you find in books and like you should learn in writing classes.) I draw attention here both to things that are very relevant for the final and areas where problems happened. Guidelines: You may opt to work on this assignment alone, with one partner, or in a group of three students. Turn in one submission per group, as always. There is to be absolutely no collaboration outside of these teams; you may not discuss the work with anyone other than your teammate(s) and the course staff. You may use any materials handed out in class and CLRS, but are strictly forbidden from using any other resources. This is a formal assignment. All of your solutions must stand alone, that is, one should not need to have this document in hand to make sense of your solutions. Present everything clearly, following standard conventions of written English and mathematical writing. Each problem should be presented starting on a new sheet of paper. We should be able to collect each problem separately without any difficulty. This assignment is due at the start of your lecture section on the last day of class, Friday, April 26, No late work is accepted; there is no time left in the semester to accept it late. Problem 1 Suppose you re running a consulting business that can move easily. It s just you and a few colleagues and a small amount of equipment. You have clients in both the northeast and northwest and are looking to optimize your business. Each month, you can either run your business out from an office in Boston (B) or in Seattle (S). The demands of your clients yield operating costs that vary based on where your current home base is. Thus, in month i, your operating cost is B i if you run the business out of Boston your operating cost is S i if you run the business out of Seattle Also, if you run the business out of one city in one month and switch to the other in the next month, you ll incur a moving cost to switch base offices. That moving cost is M. Given a sequence of n months, a plan is a sequence of n locations, each either S or B, such that the ith value represents the city where the business is housed during month i. The cost for a plan is the sum of the operating costs plus any necessary moving costs. You can begin a plan in either city. Your task is to find a optimal plan, where optimal means minimum cost in this case, given inputs M; B 1, B 2,, B n ; and S 1, S 2,, S n. For example, suppose that n = 4, M = 10, and operating costs are city month 1 month 2 month 3 month 4 Boston Seattle Then the plan of minimum cost would be {B, B, S, S} with a total cost of = 20. Your tasks: a. Show that the following strategy does not correctly solve the problem: In each month, select the city whose operating cost is smaller. b. Give and explain an example of an instance in which every optimal plan must move at least three times. Page 1 of 5 PSU CMPSC 465 Spring 2013
2 c. Derive, using the principles of dynamic programming, a function that takes the inputs given above and returns the cost for an optimal plan. As with class examples, we start with the end and consider what could happen in month n. There are two cases: We could run our consulting business in Boston in month n. We could run our consulting business in Seattle in month n. An optimal plan over n months considers information about all n months. It is impossible to make any decisions with knowing about all n months. Thus, we ultimately will need to compare the optimal value of a plan that ends in Boston with the optimal value of a plan that ends in Seattle. We will need to compute both plans using two different recursive functions. Define the following: Let OPT B (i) be the optimal cost of a plan over months 1 to i where the business runs in Boston in month i Let OPT S (i) be the optimal cost of a plan over months 1 to i where the business runs in Seattle in month i Now consider what happens if we run the business out of Boston in month i. Then it is possible that we were in Boston in the previous month or we were in Seattle in the previous month and moved. Consider each case: Case That We Were in Boston: In this case, we must pay to run in the business in Boston in month i, which incurs a cost of B i. In addition, we need to pay the cost of running the business in the first i 1 months. We want an optimal plan, and we know we were in Boston in month i 1, so the optimal cost over months 1 to i 1 is just OPT B (i 1). Case That We Were in Seattle: In this case, we must pay to run in the business in Boston in month i, which incurs a cost of B i. We also need to pay a moving cost of M to relocate the business from Seattle to Boston. Finally, we need to pay the cost of running the business in the first i 1 months. We want an optimal plan, and we know we were in Seattle in month i 1, so the optimal cost over months 1 to i 1 comes from the other optimization function: OPT S (i 1). We want the optimal cost for a plan where month i is in Boston, and optimal cost means paying less money, so we take the minimum of the costs incurred in the two cases. As B i is part of the cost in either case, though, we add it outside of the minimum calculation for simplicity. The resulting optimal cost function, defined for i 2, is: OPT B (i) = B i + min(opt B (i 1), M + OPT S (i 1)) with a base case OPT B (1) = B 1 as this base case represents running the business for one month in Boston. We must carefully define the function for running the business out of Seattle in month i. By similar reasoning, the optimal cost function, defined for i 2, is: OPT S (i) = S i + min(opt S (i 1), M + OPT B (i 1)) with a base case OPT S (1) = S 1. To find the cost of an optimal plan over all n months, then, we want the following: min(opt B (n), OPT S (n)) Some notes: 1. Some of you confused n and the index of the current month. The variable n is fixed and defined in the problem statement. 2. Some of you defined a function with two inputs and took care of the issue of where month i runs that way. Either solution is fine. 3. You must compute what happens both under the assumption we end in Boston (in month n) and under the assumption we end in Seattle (in month n). Then, you need to pick whichever of those is better, and the corresponding recurrence leads to an optimal solution (and the other becomes irrelevant). 4. Defining a month 0 with optimal costs of 0 is fine too. Page 2 of 5 PSU CMPSC 465 Spring 2013
3 d. Provide pseudocode for an efficient algorithm to compute the cost of an optimal plan. Recall from class (p. 7 of the first set of D.P. notes) that while dynamic programming solves naturally recursive problems, the subproblems would overlap were we to compute them in a top-down fashion recursively, so instead we compute them bottom-up iteratively and can get a nice Θ(n) algorithm: Let BOS_COST[1..n] be a new empty array to hold optimal costs assuming month n is in Boston Let SEA_COST[1..n] be a new empty array to hold optimal costs assuming month n is in Seattle BOS_COST[1] = B 1 SEA_COST[1] = S 1 for i = 2 to n BOS_COST[i] = B i + min(bos_cost[i 1], M + SEA_COST[i 1]) SEA_COST[i] = S i + min(sea_cost[i 1], M + BOS_COST[i 1]) return min(bos_cost[n], SEA_COST[n]) An alterative that is asymptotically as good is to compute a memoized version of the recursive algorithm. But, the simple iterative algorithm given above is definitely the cleanest solution to this problem. Page 3 of 5 PSU CMPSC 465 Spring 2013
4 Problem 2 Suppose you re managing a business that manufactures classroom materials (like chalk and whiteboard markers and paper and such) and ships them to universities all around the country. For each of the next n weeks, there is a projected supply s i of equipment (measured in pounds), which has to be shipped. Each week s supply can be carried by one of two freight companies: Company A charges a fixed rate r per pound (so it costs r s i to ship a week s supply s i ) Company B makes contracts for a fixed amount c per week, independent of how much weight is shipped. Company B only will allow contracts in blocks of three consecutive weeks at a time. Define a schedule as a sequence of freight companies over each of the n weeks and define the cost of a schedule as the total amount paid to both shipping companies over the duration of the schedule. Your tasks, on an input of a sequence of supply values s 1, s 2,, s n : a. Derive, using the principles of dynamic programming, a function that takes the inputs given above and returns the cost for an optimal schedule. Let OPT(i) be the cost of an optimal schedule over weeks 1 to i. Consider what could happen in the ith week: We could hire Company A in Week i. This means we would have to pay r s i to Company A for week i. We also need to pay for all of the prior weeks. We want to make the best schedule possible, so we ll select an optimal schedule over Weeks 1 to i 1. This incurs a cost of OPT(i 1). Thus, the grand total cost in this case is r s i + OPT(i 1). We could hire Company B in Week i. Since we must do this in three-week blocks, this means we would have had to have employed Company B in Weeks i 1 and i 2 as well. So, we d have to pay Company B for 3 weeks at a rate of c per week, or 3c total. Then we want an optimal schedule over all previous weeks, which in this case is Weeks 1 to i 3. That costs OPT(i 3). Thus, the grand total cost in this case is 3c + OPT(i 3). The best cost is certainly the cheaper, so we get, for i 3, OPT(i) = min( r s i + OPT(i 1), 3c + OPT(i 3)) Handing the base of this recurrence is much trickier. The easy case is OPT(0) = 0. But what about Weeks 1 and 2? Analysis via a possibility tree [not shown here] yields that there are three cases for Weeks 1 and 2: Pick A in both weeks. Pick A in Week 1 and B in Week 2. Pick B in both weeks. So, computing the minimum cost requires computing our recurrence using each of the three possibilities as base cases and selecting whichever gives the minimum result OPT(n). Some notes: 1. What makes this problem distinct is that it involves subproblems that go back to 1 smaller and 3 smaller. That s important. But it s not all that different from Weighted Interval Scheduling, except that now you know exactly how many weeks to look back. 2. Certainly, the trickiest part of this problem was dealing with the base cases. I won t make you do something so complicated in an exam scenario. But, you needed to be very careful here. You need to make sure you handle Weeks 1 and 2 carefully and don t define a recurrence that needs to know about Week -2 during Week 1. a. Looking at all three cases for Weeks 1 and 2 and computing the recurrence using all three possibilities is the best solution I ve seen and I ve not been convinced by anything else I saw (but there may be other tricks. ) b. You might opt to define three recurrences, one with each different set of base cases. Or, you might take the base situation as a second argument to the recurrence. 3. Related to #2, watch out for being to quick to claim that if i is 1, the optimal cost comes from picking A. It may be that this is the start of a three-week contact with B. We need to start at the end and work backwards. Page 4 of 5 PSU CMPSC 465 Spring 2013
5 b. Give and explain psuedocode for a polynomial-time algorithm that returns a schedule of minimum cost. In brief, the best solution is to define three arrays, e.g. AA_COSTS[0..n], AB_COSTS[0..n], and BB_COSTS[0..n], initialize element 0 of each to 0, and initialize elements 1 and 2 according to whichever case is selected. Then, iteratively compute elements 3 to n of each using the OPT(i) recurrence. At the end, return the min of AA_COSTS[n], AB_COSTS[n], and BB_COSTS[n]. This algorithm runs in Θ(n) time. Extra Credit Problem Implement the algorithm from Problem 1d in Java. Your program must use language-independent data structures. Also, it must follow the usual programming conventions and produce text-based output. No user interaction is required, but a test driver and text-based sample runs are. Present your work using the same template used for programming assignments. Note that you must have a correct algorithm in Problem 1d and must earn at least a B on this entire assignment to be eligible for any extra credit. (This requirement is to remind you to prioritize the important algorithm design and writing aspects of this assignment before you put any time into programming.) Disclaimer: Copying this solution and sharing it in any means outside of CMPSC 465 Spring 2013 whether with one person or posting it anywhere else is expressly forbidden. Any student who does so is in violation of the academic integrity policies of this course (and, as it is not explicitly documented, a 465 Spring 2013 student found to have shared this solution is to be penalized by a final grade drop of at least 2 letter grades) and anyone who uses this solution outside of CMPSC 465 Spring 2013 is in violation of academic integrity for wherever it is used. The bottom line: This is here to help you; please be honest and use it as it was intended only. Page 5 of 5 PSU CMPSC 465 Spring 2013
Section IV.1: Recursive Algorithms and Recursion Trees
Section IV.1: Recursive Algorithms and Recursion Trees Definition IV.1.1: A recursive algorithm is an algorithm that solves a problem by (1) reducing it to an instance of the same problem with smaller
More informationIntroduction to Algorithms March 10, 2004 Massachusetts Institute of Technology Professors Erik Demaine and Shafi Goldwasser Quiz 1.
Introduction to Algorithms March 10, 2004 Massachusetts Institute of Technology 6.046J/18.410J Professors Erik Demaine and Shafi Goldwasser Quiz 1 Quiz 1 Do not open this quiz booklet until you are directed
More informationBinary Search Trees CMPSC 122
Binary Search Trees CMPSC 122 Note: This notes packet has significant overlap with the first set of trees notes I do in CMPSC 360, but goes into much greater depth on turning BSTs into pseudocode than
More informationDynamic Programming. Lecture 11. 11.1 Overview. 11.2 Introduction
Lecture 11 Dynamic Programming 11.1 Overview Dynamic Programming is a powerful technique that allows one to solve many different types of problems in time O(n 2 ) or O(n 3 ) for which a naive approach
More informationCS 2302 Data Structures Spring 2015
1. General Information Instructor: CS 2302 Data Structures Spring 2015 Olac Fuentes Email: ofuentes@utep.edu Web: www.cs.utep.edu/ofuentes Office hours: Tuesdays and Thursdays 2:00-3:30, or by appointment,
More informationKenken For Teachers. Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles June 27, 2010. Abstract
Kenken For Teachers Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles June 7, 00 Abstract Kenken is a puzzle whose solution requires a combination of logic and simple arithmetic skills.
More informationOptimal Binary Search Trees Meet Object Oriented Programming
Optimal Binary Search Trees Meet Object Oriented Programming Stuart Hansen and Lester I. McCann Computer Science Department University of Wisconsin Parkside Kenosha, WI 53141 {hansen,mccann}@cs.uwp.edu
More informationOrder of Operations More Essential Practice
Order of Operations More Essential Practice We will be simplifying expressions using the order of operations in this section. Automatic Skill: Order of operations needs to become an automatic skill. Failure
More informationRecursive Algorithms. Recursion. Motivating Example Factorial Recall the factorial function. { 1 if n = 1 n! = n (n 1)! if n > 1
Recursion Slides by Christopher M Bourke Instructor: Berthe Y Choueiry Fall 007 Computer Science & Engineering 35 Introduction to Discrete Mathematics Sections 71-7 of Rosen cse35@cseunledu Recursive Algorithms
More informationMcKinsey Problem Solving Test Top Tips
McKinsey Problem Solving Test Top Tips 1 McKinsey Problem Solving Test You re probably reading this because you ve been invited to take the McKinsey Problem Solving Test. Don t stress out as part of the
More informationThe Tower of Hanoi. Recursion Solution. Recursive Function. Time Complexity. Recursive Thinking. Why Recursion? n! = n* (n-1)!
The Tower of Hanoi Recursion Solution recursion recursion recursion Recursive Thinking: ignore everything but the bottom disk. 1 2 Recursive Function Time Complexity Hanoi (n, src, dest, temp): If (n >
More informationPrice Theory Lecture 4: Production & Cost
Price Theory Lecture 4: Production & Cost Now that we ve explained the demand side of the market, our goal is to develop a greater understanding of the supply side. Ultimately, we want to use a theory
More informationLecture 2 Mathcad Basics
Operators Lecture 2 Mathcad Basics + Addition, - Subtraction, * Multiplication, / Division, ^ Power ( ) Specify evaluation order Order of Operations ( ) ^ highest level, first priority * / next priority
More informationCSIS 202: Introduction to Computer Science Spring term 2015-2016 Midterm Exam
Page 0 German University in Cairo April 7, 2016 Media Engineering and Technology Prof. Dr. Slim Abdennadher Dr. Hisham Othman CSIS 202: Introduction to Computer Science Spring term 2015-2016 Midterm Exam
More informationMathematics Task Arcs
Overview of Mathematics Task Arcs: Mathematics Task Arcs A task arc is a set of related lessons which consists of eight tasks and their associated lesson guides. The lessons are focused on a small number
More informationCSE373: Data Structures and Algorithms Lecture 1: Introduction; ADTs; Stacks/Queues. Linda Shapiro Spring 2016
CSE373: Data Structures and Algorithms Lecture 1: Introduction; ADTs; Stacks/Queues Linda Shapiro Registration We have 180 students registered and others who want to get in. If you re thinking of dropping
More informationCS104: Data Structures and Object-Oriented Design (Fall 2013) October 24, 2013: Priority Queues Scribes: CS 104 Teaching Team
CS104: Data Structures and Object-Oriented Design (Fall 2013) October 24, 2013: Priority Queues Scribes: CS 104 Teaching Team Lecture Summary In this lecture, we learned about the ADT Priority Queue. A
More informationLecture 13: The Knapsack Problem
Lecture 13: The Knapsack Problem Outline of this Lecture Introduction of the 0-1 Knapsack Problem. A dynamic programming solution to this problem. 1 0-1 Knapsack Problem Informal Description: We have computed
More informationMAKE BIG MONEY QUICKLY! Low Start Up Cost! Easy To Operate Business! UNLIMITED INCOME POTENTIAL!
MAKE BIG MONEY QUICKLY! Low Start Up Cost! Easy To Operate Business! UNLIMITED INCOME POTENTIAL! In this incredible $12 BILLION Dollar Industry You Can Join Today and Start Making Serious Profits in as
More informationCOMP 250 Fall 2012 lecture 2 binary representations Sept. 11, 2012
Binary numbers The reason humans represent numbers using decimal (the ten digits from 0,1,... 9) is that we have ten fingers. There is no other reason than that. There is nothing special otherwise about
More information3. Mathematical Induction
3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)
More informationSTART HERE THE BASICS TIPS + TRICKS ADDITIONAL HELP. quick start THREE SIMPLE STEPS TO SET UP IN UNDER 5 MINUTES
quick start Thank you for choosing Virtual Wallet! We hope Virtual Wallet will help you spend, save and grow your money. This Quick Start will introduce you to some of Virtual Wallet s features so you
More informationA guide to Sage One Accounts from your accountant
SageOne Accounts A guide to Sage One Accounts from your accountant About Sage One Sage One is a series of online services for small business owners and their accountants, allowing them to manage their
More informationMake and register your lasting power of attorney a guide
LP12 Make and register your lasting power of attorney a guide Financial decisions including: running your bank and savings accounts making or selling investments paying your bills buying or selling your
More informationGrade 7/8 Math Circles Sequences and Series
Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 7/8 Math Circles Sequences and Series November 30, 2012 What are sequences? A sequence is an ordered
More informationHelping you find and get on in work
welcome guide CONTENTS Introduction What Universal Credit is, and what you ll need to do to claim it. This section tells you what you ll get if you re doing all you can to find work, and what will happen
More information16. Recursion. COMP 110 Prasun Dewan 1. Developing a Recursive Solution
16. Recursion COMP 110 Prasun Dewan 1 Loops are one mechanism for making a program execute a statement a variable number of times. Recursion offers an alternative mechanism, considered by many to be more
More informationPre-Algebra Lecture 6
Pre-Algebra Lecture 6 Today we will discuss Decimals and Percentages. Outline: 1. Decimals 2. Ordering Decimals 3. Rounding Decimals 4. Adding and subtracting Decimals 5. Multiplying and Dividing Decimals
More informationPigeonhole Principle Solutions
Pigeonhole Principle Solutions 1. Show that if we take n + 1 numbers from the set {1, 2,..., 2n}, then some pair of numbers will have no factors in common. Solution: Note that consecutive numbers (such
More informationHOW TO GET THE MOST OUT OF YOUR ACCOUNTANT
Prepared by HOW TO GET THE MOST OUT OF YOUR ACCOUNTANT Do you find dealing with your accountant confusing? Not exactly sure what services an accountant can provide for you? Do you have expensive accounting
More informationEquity Release your essential guide
Equity Release your essential guide Welcome This guide has been put together to explain equity release, what it means and the options it can offer. We aim to give you as broad an overview as possible and
More informationHandouts for teachers
ASKING QUESTIONS THAT ENCOURAGE INQUIRY- BASED LEARNING How do we ask questions to develop scientific thinking and reasoning? Handouts for teachers Contents 1. Thinking about why we ask questions... 1
More informationSequential Data Structures
Sequential Data Structures In this lecture we introduce the basic data structures for storing sequences of objects. These data structures are based on arrays and linked lists, which you met in first year
More informationc. Given your answer in part (b), what do you anticipate will happen in this market in the long-run?
Perfect Competition Questions Question 1 Suppose there is a perfectly competitive industry where all the firms are identical with identical cost curves. Furthermore, suppose that a representative firm
More informationNPI Number Everything You Need to Know About NPI Numbers
NPI Number Everything You Need to Know About NPI Numbers By Alice Scott and Michele Redmond What is an NPI number, who needs one, how do you get one, when do you need two NPI # s, what is a taxonomy code
More informationARKANSAS WORKERS COMPENSATION COMMISSION 324 Spring Street P.O. Box 950 Little Rock, AR 72203-0950
ARKANSAS WORKERS COMPENSATION COMMISSION 324 Spring Street P.O. Box 950 Little Rock, AR 72203-0950 TO: FROM: Interested Parties Carl Bayne Operations/Compliance DATE: November 20, 2012 SUBJECT: Form 2
More informationBasic Components of an LP:
1 Linear Programming Optimization is an important and fascinating area of management science and operations research. It helps to do less work, but gain more. Linear programming (LP) is a central topic
More informationData Structures and Algorithms Written Examination
Data Structures and Algorithms Written Examination 22 February 2013 FIRST NAME STUDENT NUMBER LAST NAME SIGNATURE Instructions for students: Write First Name, Last Name, Student Number and Signature where
More informationInvestigating Investment Formulas Using Recursion Grade 11
Ohio Standards Connection Patterns, Functions and Algebra Benchmark C Use recursive functions to model and solve problems; e.g., home mortgages, annuities. Indicator 1 Identify and describe problem situations
More informationNear Optimal Solutions
Near Optimal Solutions Many important optimization problems are lacking efficient solutions. NP-Complete problems unlikely to have polynomial time solutions. Good heuristics important for such problems.
More informationIf you buy insurance, you are making a personal decision about how you want to manage your risk if things go wrong.
http://understandinsurance.com.au/buying-insurance Buying insurance If you buy insurance, you are making a personal decision about how you want to manage your risk if things go wrong. There are a large
More informationUmmmm! Definitely interested. She took the pen and pad out of my hand and constructed a third one for herself:
Sum of Cubes Jo was supposed to be studying for her grade 12 physics test, but her soul was wandering. Show me something fun, she said. Well I wasn t sure just what she had in mind, but it happened that
More informationFinancial Mathematics
Financial Mathematics For the next few weeks we will study the mathematics of finance. Apart from basic arithmetic, financial mathematics is probably the most practical math you will learn. practical in
More informationSession 6 Number Theory
Key Terms in This Session Session 6 Number Theory Previously Introduced counting numbers factor factor tree prime number New in This Session composite number greatest common factor least common multiple
More informationCPS122 - OBJECT-ORIENTED SOFTWARE DEVELOPMENT. Team Project
CPS122 - OBJECT-ORIENTED SOFTWARE DEVELOPMENT Team Project Due Dates: See syllabus for due dates for each milestone This project spans much of the semester, to be completed as a series of milestones, each
More informationKEY FEATURES OF THE PERSONAL PENSION (TOP UP PLAN) Important information you need to read
KEY FEATURES OF THE PERSONAL PENSION (TOP UP PLAN) Important information you need to read THE FINANCIAL CONDUCT AUTHORITY IS A FINANCIAL SERVICES REGULATOR. IT REQUIRES US, SCOTTISH WIDOWS, TO GIVE YOU
More informationPT AVENUE GUIDE OVERVIEW
PT AVENUE GUIDE OVERVIEW WSPTA is currently undertaking a database conversion from imis (the previous membership database) to a cloud based service called PT Avenue. The primary reason for this conversion
More informationQuick Start Guide Getting Started with Stocks
Quick Start Guide Getting Started with Stocks Simple but Sophisticated Don t let the name fool you: the scan may be simple, but behind the curtain is a very sophisticated process designed to bring you
More informationWhat you should know about: Windows 7. What s changed? Why does it matter to me? Do I have to upgrade? Tim Wakeling
What you should know about: Windows 7 What s changed? Why does it matter to me? Do I have to upgrade? Tim Wakeling Contents What s all the fuss about?...1 Different Editions...2 Features...4 Should you
More informationWRITING PROOFS. Christopher Heil Georgia Institute of Technology
WRITING PROOFS Christopher Heil Georgia Institute of Technology A theorem is just a statement of fact A proof of the theorem is a logical explanation of why the theorem is true Many theorems have this
More informationThe Mathematics 11 Competency Test Percent Increase or Decrease
The Mathematics 11 Competency Test Percent Increase or Decrease The language of percent is frequently used to indicate the relative degree to which some quantity changes. So, we often speak of percent
More informationExercise 8: SRS - Student Registration System
You are required to develop an automated Student Registration System (SRS). This system will enable students to register online for courses each semester. As part of the exercise you will have to perform
More informationThe Cost of Production
The Cost of Production 1. Opportunity Costs 2. Economic Costs versus Accounting Costs 3. All Sorts of Different Kinds of Costs 4. Cost in the Short Run 5. Cost in the Long Run 6. Cost Minimization 7. The
More informationMOST FREQUENTLY ASKED INTERVIEW QUESTIONS. 1. Why don t you tell me about yourself? 2. Why should I hire you?
MOST FREQUENTLY ASKED INTERVIEW QUESTIONS 1. Why don t you tell me about yourself? The interviewer does not want to know your life history! He or she wants you to tell how your background relates to doing
More informationThis booklet is for candidates who are applying for entry level jobs with New York State and with local governments in the state.
New York State Department of Civil Service Publication Civil Service Examinations How To Take A Written Test This booklet is for candidates who are applying for entry level jobs with New York State and
More informationChapter 11 Number Theory
Chapter 11 Number Theory Number theory is one of the oldest branches of mathematics. For many years people who studied number theory delighted in its pure nature because there were few practical applications
More informationSouth Dakota. Opportunity Scholarship. Frequently Asked Questions
South Dakota Opportunity Scholarship Frequently Asked Questions Question: What is the difference between a Regents Scholar Diploma and the South Dakota Opportunity Scholarship? Answer: The Regents Scholar
More informationLecture 12 Doubly Linked Lists (with Recursion)
Lecture 12 Doubly Linked Lists (with Recursion) In this lecture Introduction to Doubly linked lists What is recursion? Designing a node of a DLL Recursion and Linked Lists o Finding a node in a LL (recursively)
More informationSession 7 Fractions and Decimals
Key Terms in This Session Session 7 Fractions and Decimals Previously Introduced prime number rational numbers New in This Session period repeating decimal terminating decimal Introduction In this session,
More informationDiscrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 10
CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 10 Introduction to Discrete Probability Probability theory has its origins in gambling analyzing card games, dice,
More informationTo be used in conjunction with the Invitation to Tender for Consultancy template.
GUIDANCE NOTE Tendering for, choosing and managing a consultant Using this guidance This information is not intended to be prescriptive, but for guidance only. Appointing consultants for relatively small
More informationBetter Together. Best regards, Team Gynzy
www.gynzy.com Better Together As a teacher who is in the classroom all day, you know much better than we do about the needs of students and teachers. Which is why we strive to continuously improve Gynzy
More information6th Grade Lesson Plan: Probably Probability
6th Grade Lesson Plan: Probably Probability Overview This series of lessons was designed to meet the needs of gifted children for extension beyond the standard curriculum with the greatest ease of use
More informationCASH ISA SAVINGS CONDITIONS. For use from 2nd September 2016.
CASH ISA SAVINGS CONDITIONS. For use from 2nd September 2016. WELCOME TO HALIFAX This booklet explains how your Halifax savings account works, and includes its main conditions. This booklet contains: information
More informationDemand, Supply, and Market Equilibrium
3 Demand, Supply, and Market Equilibrium The price of vanilla is bouncing. A kilogram (2.2 pounds) of vanilla beans sold for $50 in 2000, but by 2003 the price had risen to $500 per kilogram. The price
More informationPROBLEM SOLVING. 1. I m thinking of buying to let - where do I start?
Top Buy to Let FAQs 1. I m thinking of buying to let - where do I start? If you re not sure whether buy to let is right for you or want to make sure you buy a property that ll give you the returns you
More informationModels for Incorporating Block Scheduling in Blood Drive Staffing Problems
University of Arkansas, Fayetteville ScholarWorks@UARK Industrial Engineering Undergraduate Honors Theses Industrial Engineering 5-2014 Models for Incorporating Block Scheduling in Blood Drive Staffing
More informationUnderstanding Options: Calls and Puts
2 Understanding Options: Calls and Puts Important: in their simplest forms, options trades sound like, and are, very high risk investments. If reading about options makes you think they are too risky for
More informationWhat Is Recursion? Recursion. Binary search example postponed to end of lecture
Recursion Binary search example postponed to end of lecture What Is Recursion? Recursive call A method call in which the method being called is the same as the one making the call Direct recursion Recursion
More informationUnit 5 Length. Year 4. Five daily lessons. Autumn term Unit Objectives. Link Objectives
Unit 5 Length Five daily lessons Year 4 Autumn term Unit Objectives Year 4 Suggest suitable units and measuring equipment to Page 92 estimate or measure length. Use read and write standard metric units
More informationNAME OF ASSESSMENT: Reading Informational Texts and Opinion Writing Performance Assessment
GRADE: Third Grade NAME OF ASSESSMENT: Reading Informational Texts and Opinion Writing Performance Assessment STANDARDS ASSESSED: Students will ask and answer questions to demonstrate understanding of
More informationReading 13 : Finite State Automata and Regular Expressions
CS/Math 24: Introduction to Discrete Mathematics Fall 25 Reading 3 : Finite State Automata and Regular Expressions Instructors: Beck Hasti, Gautam Prakriya In this reading we study a mathematical model
More informationWhat qualities are employers looking for in teen workers? How can you prove your own skills?
Sell Yourself 4 Finding a job The BIG Idea What qualities are employers looking for in teen workers? How can you prove your own skills? AGENDA Approx. 45 minutes I. Warm Up: Employer Survey Review (15
More informationResearch Tools & Techniques
Research Tools & Techniques for Computer Engineering Ron Sass http://www.rcs.uncc.edu/ rsass University of North Carolina at Charlotte Fall 2009 1/ 106 Overview of Research Tools & Techniques Course What
More information2 Mathematics Curriculum
New York State Common Core 2 Mathematics Curriculum GRADE GRADE 2 MODULE 3 Topic G: Use Place Value Understanding to Find 1, 10, and 100 More or Less than a Number 2.NBT.2, 2.NBT.8, 2.OA.1 Focus Standard:
More information14.1 Rent-or-buy problem
CS787: Advanced Algorithms Lecture 14: Online algorithms We now shift focus to a different kind of algorithmic problem where we need to perform some optimization without knowing the input in advance. Algorithms
More informationRoman Numerals Case Study 1996 M. J. Clancy and M. C. Linn
Roman Numerals Case Study 1996 M. J. Clancy and M. C. Linn Background Values less than 3999 in the Roman numeral system are written using seven digits whose decimal equivalents are given in the table below.
More informationScheduling. Getting Started. Scheduling 79
Scheduling 9 Scheduling An event planner has to juggle many workers completing different tasks, some of which must be completed before others can begin. For example, the banquet tables would need to be
More informationData Structures and Algorithms
Data Structures and Algorithms CS245-2016S-06 Binary Search Trees David Galles Department of Computer Science University of San Francisco 06-0: Ordered List ADT Operations: Insert an element in the list
More informationBT Cloud Phone. A simpler way to manage your business calls.
. A simpler way to manage your business calls. Make and receive calls from anywhere with a phone system that s hosted in the cloud. Save on set-up, maintenance and call charges. Have reassurance that we
More informationAbstract The purpose of this paper is to present the results of my action research which was conducted in several 7 th /8 th grade language arts
Abstract The purpose of this paper is to present the results of my action research which was conducted in several 7 th /8 th grade language arts class periods in a Spanish immersion program over a two
More informationHow to Make the Most of Excel Spreadsheets
How to Make the Most of Excel Spreadsheets Analyzing data is often easier when it s in an Excel spreadsheet rather than a PDF for example, you can filter to view just a particular grade, sort to view which
More informationKey features of the Home Retail Group Personal Pension Plan
Key features of the Home Retail Group Personal Pension Plan This is an important document which you should keep in a safe place. You may need to read it in future. Home Retail Group Personal Pension Plan
More informationUsing trusts can help to make sure your financial plans take care of the future
Using trusts can help to make sure your financial plans take care of the future 2 Trusts and what they do If you ve already taken the time to make financial plans for the future, using trusts can help
More informationOnline Fixed Rate Cash ISA Range
Online Fixed Rate Range This document provides you with key information about s so that you can make an informed and confident choice about saving with a Skipton. s protect your savings interest from Personal
More informations = 1 + 2 +... + 49 + 50 s = 50 + 49 +... + 2 + 1 2s = 51 + 51 +... + 51 + 51 50 51. 2
1. Use Euler s trick to find the sum 1 + 2 + 3 + 4 + + 49 + 50. s = 1 + 2 +... + 49 + 50 s = 50 + 49 +... + 2 + 1 2s = 51 + 51 +... + 51 + 51 Thus, 2s = 50 51. Therefore, s = 50 51. 2 2. Consider the sequence
More informationProblem Set 7 Solutions
8 8 Introduction to Algorithms May 7, 2004 Massachusetts Institute of Technology 6.046J/18.410J Professors Erik Demaine and Shafi Goldwasser Handout 25 Problem Set 7 Solutions This problem set is due in
More informationFirst programming project: Key-Word indexer
University of Illinois at Chicago CS 202: Data Structures and Discrete Mathematics II Handout 2 Professor Robert H. Sloan Friday, August 30, 2002 First programming project: Key-Word indexer 1 Indexing
More information1.2 Solving a System of Linear Equations
1.. SOLVING A SYSTEM OF LINEAR EQUATIONS 1. Solving a System of Linear Equations 1..1 Simple Systems - Basic De nitions As noticed above, the general form of a linear system of m equations in n variables
More informationA Guide to Cover Letter Writing
A Guide to Cover Letter Writing Contents What is a Cover Letter?... 2 Before you get started - Do your Research... 3 Formatting the letter... 4 Cover letter content... 5 Section 1 - Opening... 5 Section
More informationA Quick Algebra Review
1. Simplifying Epressions. Solving Equations 3. Problem Solving 4. Inequalities 5. Absolute Values 6. Linear Equations 7. Systems of Equations 8. Laws of Eponents 9. Quadratics 10. Rationals 11. Radicals
More information15-150 Lecture 11: Tail Recursion; Continuations
15-150 Lecture 11: Tail Recursion; Continuations Lecture by Dan Licata February 21, 2011 In this lecture we will discuss space usage: analyzing the memory it takes your program to run tail calls and tail
More informationSample Online Syllabus
Sample Online Syllabus This sample syllabus is based off of the MCCC DE-1 form and is designed to provide suggestions to instructors seeking to create a comprehensive syllabus for teaching online. Please
More informationBUSINESS OCR LEVEL 2 CAMBRIDGE TECHNICAL. Cambridge TECHNICALS FINANCIAL FORECASTING FOR BUSINESS CERTIFICATE/DIPLOMA IN K/502/5252 LEVEL 2 UNIT 3
Cambridge TECHNICALS OCR LEVEL 2 CAMBRIDGE TECHNICAL CERTIFICATE/DIPLOMA IN BUSINESS FINANCIAL FORECASTING FOR BUSINESS K/502/5252 LEVEL 2 UNIT 3 GUIDED LEARNING HOURS: 30 UNIT CREDIT VALUE: 5 FINANCIAL
More informationSocial Return on Investment
Social Return on Investment Valuing what you do Guidance on understanding and completing the Social Return on Investment toolkit for your organisation 60838 SROI v2.indd 1 07/03/2013 16:50 60838 SROI v2.indd
More informationScheduling Best Practices
RECOVERY SCHEDULES By Mark I. Anderson, Executive Vice President and Chief Operating Officer of Warner Construction Consultants, Inc. In this article, as part of a continuing series of articles regarding
More informationSolutions to Math 51 First Exam January 29, 2015
Solutions to Math 5 First Exam January 29, 25. ( points) (a) Complete the following sentence: A set of vectors {v,..., v k } is defined to be linearly dependent if (2 points) there exist c,... c k R, not
More information12 Tips for Negotiating with Suppliers
12 Tips for Negotiating with Suppliers Written by: Lisa Suttora This workbook is not a free ebook. If you would like to recommend this product to others, please do so at www.workwithsuppliers.com/businesssetup
More informationCriminal Justice I. Mr. Concannon Smith Email: Benjamin_Smith@wrsd.net Website: www.benjaminallensmith.com Twitter: @BACSmith
Criminal Justice I Mr. Concannon Smith Email: Benjamin_Smith@wrsd.net Website: www.benjaminallensmith.com Twitter: @BACSmith Course Description: This course has two goals. The first is to provide students
More informationHow to Use Solo Ads to. Grow Your Business
How to Use Solo Ads to Grow Your Business By: Reed Floren http://www.soloaddirectory.com The Basics of Solo Ads Solo ads a great way to market your business using a small scale method. Solo ads are basically
More information